91Ó°ÊÓ

In circular motion, tangential acceleration refers to how fast an object's speed is changing along the path of the circle. It is directly related to the net tangential force acting on the object. According to Newton's second law, the relationship between force, mass, and acceleration is described with the formula:

• Net force = mass × acceleration

For this speedboat example, with a given force of 550 N and a mass of 220 kg, the tangential acceleration is calculated by rearranging the formula:

• \( a_t = \frac{F_t}{m} \)

By substituting the values:

• \( a_t = \frac{550 \, \text{N}}{220 \, \text{kg}} = 2.5 \, \text{m/s}^2 \)

This indicates that the speed of the boat increases by 2.5 meters per second every second due to the engine's force during the circular turn. It's important to understand how tangential force and mass work together to result in this acceleration.

Centripetal acceleration is a crucial concept in circular motion. It is the acceleration that keeps an object moving in a circular path, always directed toward the center of the circle. In our speedboat problem, after 2 seconds in the turn, the boat's speed increases from 5.0 m/s to 10.0 m/s. This new speed is used to calculate the centripetal acceleration.
The formula is:

• \( a_c = \frac{v^2}{r} \)

Here, \( v \) is the tangential speed after 2 seconds, and \( r \) is the radius of the circular path (32 m). Substituting the known values:

• \( a_c = \frac{(10.0 \, \text{m/s})^2}{32 \, \text{m}} = 3.125 \, \text{m/s}^2 \)

Remember, centripetal acceleration doesn't change the speed of the object, but changes the direction of the object's velocity, ensuring it continues along a circular path.

Newton's second law of motion is a fundamental principle that links force, mass, and acceleration. In simple terms, it states:

• A net force acting on a body produces an acceleration that is inversely proportional to its mass and directly proportional to the force.

This law is described by the equation:

• \( F = ma \)

Where:

• \( F \) is the force acting on the object,
• \( m \) is the object's mass, and
• \( a \) is the resulting acceleration.

In our problem, we specifically applied Newton's second law to determine tangential acceleration. With the net tangential force known, calculating the acceleration involved simply dividing this force by the boat's mass:

• \( a_t = \frac{550 \, \text{N}}{220 \, \text{kg}} = 2.5 \, \text{m/s}^2 \)

This illustrates how changes in either force or mass can profoundly impact acceleration, highlighting an essential aspect of dynamics in circular motion. Newton's second law is pivotal for understanding the mechanics behind varied acceleration types, such as both tangential and centripetal seen in our problem.